Solving the Taylor problem with horizontal viscosity

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Solving the Taylor

problem with horizontal

viscosity

Pieter C. Roos

Water Engineering & Management, University of Twente

Henk M. Schuttelaars

Delft Institutie of Applied Mathematics, TU Delft

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1. Motivation and goal

• Understand morphodynamics of tidal basins • Tool: process-based model for tidal flow

– Smooth flow field required  add horizontal viscosity

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2. Background: inviscid

Taylor problem

• Semi-infinite rectangular basin of uniform depth – No-normal flow BC – Inviscid shallow water eqs. – Incoming Kelvin wave

Co-tidal and co-range chart

Tidal current ellipses

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2. Background: inviscid

Taylor problem

• Semi-infinite rectangular basin of uniform depth • Solution as superposition of ‘open channel modes’

– Kelvin & Poincaré waves

– Collocation method – Amphidromic system

and tidal current ellipses

Co-tidal and co-range chart

Tidal current ellipses

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2. Extending inviscid Taylor

problem…

• Semi-infinite rectangular basin of uniform depth • Solution as superposition of

‘open channel modes’ • Extension to arbitrary

box-type geometries

– Problems for flow field at reflex angle-corners – Remedy: add viscosity

Tidal current ellipses_|u|(x,y)

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3. Viscous Taylor

problem

• Geometry and boundary conditions

– Free surface elevation ζ, depth-averaged flow (u,v)

– No slip at closed boundaries: (u,v)=0

– Incoming Kelvin wave from x=+∞

x=0 B x→ y↑ Kelvin wave Uniform depth H

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3. Viscous Taylor

problem

• Geometry and boundary conditions

• Linearized shallow water equations –

at

O(Fr

0

)

gζ_x + u_t– fv = ν[u_xx+u_yy] gζy + vt + fu = ν[vxx+vyy]

ζt + [Hu]x + [Hv]y = 0

– Acceleration of gravity g, Coriolis parameter f,

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3. Viscous Taylor

problem

• Geometry and boundary conditions

• Linearized shallow water equations –

at O(Fr

0

)

• Solution method

– Find viscous ‘open channel modes’

– Write solution as a superposition of these modes

– Use collocation method to satisfy no slip BC at x=0

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4. Results: open channel

modes

• General form: ζ(x,y,t) = Z(y)exp(i[ωt-kx])

+ c.c.

– Angular frequency ω, (complex) wave number k

– Transverse structure:

Z(y) = Z₁e-αy + Z₂e-βy + Z₃eα[y-B] + Z₄eβ[y-B]

– Solvability condition from BCs at y=0,B 

k, α, β, Z_j x=0 B x→ y↑ Uniform depth H

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4. Open channel

modes

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4. Open channel

modes

inviscid viscous

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4. Open channel

modes

• Viscous Kelvin and Poincaré

modes

– Boundary layers at y=0,B

– Interior structure similar to inviscid case

– Viscous dissipation, slight decrease in length scales

x=0 B x→ y↑ Uniform depth H viscous

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4. Viscous Kelvin

mode

ζ(x,y,t) u(x,y,t) v(x,y,t) viscous

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4. Viscous

Poincaré modes

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4. Viscous

Poincaré modes

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4. New modes

viscous ζ(x,y,t) u(x,y,t) v(x,y,t)

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4. Viscous Taylor solution

• Truncated superposition of open

channel modes

– Incoming Kelvin wave and 2N+1 reflected modes

• Use collocation method to satisfy

no-slip BC at x=0

– N+1 points where u=0 and N points where v=0 x=0 x→ y↑ Kelvin wave v=0 u=0

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4. Viscous Taylor

solution

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5. Conclusions

• Taylor problem has been extended to

account for horizontally viscous

effects

– No-slip condition at closed boundaries

• Solution involves viscous open

channel modes

– Viscous Kelvin and Poincaré modes

– A new type of mode arises, responsible for the transverse boundary layer at x=0

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6. Outlook

• Details of collocation method

• Residual flow and higher harmonics

– Nonlinear M2-interactions at O(Fr1)  M0,

M4

• Geometrical extension of viscous model

– To arbitrary box-type geometries  smooth flow field

– Applications: artificial islands, inlets, obstructions

Solving the Taylor problem with horizontal viscosity